The CEO of Mount Sinai Health System in New York City sees hospitals becoming a one-stop shop for getting care and the insurance to pay for it. That's "the future," Dr. Kenneth Davis told CNBC on Wednesday.

"Part of the motivation for us to become a much larger hospital system—bigger geographic area—was to be able to accept risk," he said in a "Squawk Box" interview.

Another major New York hospital organization has already taken the step to provide health insurance. The day before the Oct. 1 launch of the Obamacare online exchanges, North Shore-LIJ Health System CEO Michael Dowling told "Squawk Box" that new health insurance from his group would be part of the New York state marketplace.

New York—like 13 other states and the District of Columbia—runs its own exchange. Residents from 36 other states have to buy health insurance through the federally operated Healthcare.gov website, which has been plagued with technology issues.

Government officials and outside contractors are racing to fix the problems before Dec. 15—the last day for people to sign-up for coverage to begin on Jan. 1. The Obama administration has set Nov. 30 as its own deadline for getting everything running smoothly.

Tech glitches aside, Davis speculated on why HMOs failed and Obamacare might not. "The reason is the HMOs coordinated care by the insurance companies. What's happened recently, whether it's from the insurance companies or Obamacare, is that the care people—the doctors—are empowered to coordinate that care."

"When you put the insurance company between you and your care you begin to think, 'Well, their interest is for me to get less and less care. They don't know anything about me,'" Davis said. "But when it's your doctor telling you wait a minute, 'You don't need that care,'" that's when patients start to trust the system and better outcomes are achieved.

Before leading the hospital, Davis conducted groundbreaking work on Alzheimer's research as part of his specialty in geriatric psychiatry.